Adaptive Multiresolution Collocation Methods for Initial-Boundary Value Problems of Nonlinear PDEs

Abstract

We have designed a cubic spline wavelet-like decomposition for the Sobolev space H (I) where I is a bounded interval. Based on a special point value vanishing property of the wavelet basis functions, a fast discrete wavelet transform (DWT) is constructed. This DWT will map discrete samples of a function to its wavelet expansion coefficients in at most 7N log N operations. Using this transform, we propose a collocation method for the initial boundary value problem of nonlinear partial differential equations (PDEs). Then, we test the efficiency of the DWT and apply the collocation method to solve linear and nonlinear PDEs.

Neuronal Processes Involved in Subjective Feeling Emergence: Oscillatory Activity During an Emotional Monitoring Task

“…Prior research has shown that different frequency bands are sensitive to different tasks. In particular, increases in oscillations in gamma and beta bands (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) have been reported during complex cognitive functions (mostly attention, learning, and memory, [14,15]). Gamma oscillations have been implicated in the creation of representation and consciousness emergence (see also [16], and beta oscillations have been indirectly implicated in feature binding [17].…”

An efficient nonlinear circuit simulation technique

“…However, for strongly nonlinear circuits, the use of harmonic balance for the inner loop can prove limited. Another approach for solving (3) using a pseudowavelet collocation method derived from that proposed by Cai and Wang [10] has been suggested by the authors [9]. In Section III, a brief overview of the wavelet collocation scheme is given.…”

“…where and are scaling and wavelet functions, respectively, and are spline functions introduced to approximate boundary nonhomogeneities, as described in [10]. are the unknown coefficients, which are a function of only.…”

“…As proposed by Condon and Dautbegović in [9], (3) is then collocated on collocation points in to result in a semidiscretized equation system. The interpolation points are those as chosen in [10]. At this juncture, a nonlinear model reduction strategy is employed.…”

“…A modification of the wavelet-based collocation approach proposed by Cai and Wang in [10] forms the core of the technique and, unlike Christoffersen and Steer [11], the cubic spline wavelet basis is employed to solve the multitime partial differential-equation representation of the system rather than the original ordinary differential-equation representation. However, the technique presented in [9] is greatly enhanced in this paper, yielding considerable gains in efficiency in two respects.…”